The concepts of Mandelbrot's fractal geometry have been applied to the structure of individual central nervous system neurons and other cell types grown in tissue culture or from whole animals. Techniques have been developed to measure the "fractal dimension" (FD), which is a measure of complexity of individual cells' structure, with particular reference to the degree of their dendritic branching and the roundness of their borders. These techniques were calibrated against images of known FD and then employed to measure the unknown FD of individual neurons. The range of FD's in 28 neurons examined was between 1.14, indicating a relatively low complexity, to 1.60, indicating a high complexity. The values obtained not only agreed with the appearance of the neurons to trained investigators but also correlated (C.C. =0.68) with another measure of image complexity (Aspect Ratio = perimeter2/area). In addition, we have begun applying our methods to other cell types (e.g., retinal neurons, neuroblastoma cells, glia, etc.)